WEIGHTED INTEGRABILITY RESULTS FOR FIRST HANKEL-CLIFFORD TRANSFORM Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 12 (30), No 2, 2023, pp. 107-117

DOI: 10.15393/j3.art.2023.13050

107

UDC 517.544

S. S. VOLOSIVETS

WEIGHTED INTEGRABILITY RESULTS FOR FIRST HANKEL-CLIFFORD TRANSFORM

Abstract. We obtain sufficient conditions for the weighted inte-grability of the first Hankel-Clifford transforms of functions from generalized integral Lipschitz classes. These conditions are analogues and generalization of well-known Titchmarsh conditions for the classical Fourier transform.

Key words: first Hankel-Clifford transform, Hankel-Clifford translation, generalized Lipschitz spaces, weighted integrability

2020 Mathematical Subject Classification: 44Á15, 47Á10

1. Introduction. Let f: R ñ C be an integrable function in Lebesgue's sense over R (f e L1(R)). Then the Fourier transform of f is defined by

f(x) = (2n)-1/2 j f (t)e~itx dt, x e R.

R

In the case 1 < p ^ 2, the Fourier transform of a function f e LP(R) is

b

defined as the limit of (2^)~1{2 $ f (x)e~ttx dx in the norm of Lq(R), where

a

1/p + 1/q = 1 and a ñ —8, b ñ +8.

In particular, f e Lq(R) and the following Hausdorff-Young inequality:

\\f\\q ^ C\\f ||p := C ^\f (t)\p dt)1/P, f e LP(R), 1 < p ^ 2, (1)

R

holds. For p = 2, the inequality in (1) is substituted by the Plancherel equality. More about these results can be found in [19, Ch. III and IV] or [3, Ch. 5].

© Petrozavodsk State University, 2023

For f e LP(R), 1 ^ p < 8, we consider the modulus of smoothness of order k e N

uk(t,S)p = sup \\Akhf \\p, Akhf (x) = ]](—1)'Yk)f (x + (k — 2j)h/2). o^MJ J"0 \3J

The following result of Titchmarsh is well known (see [19, Ch. 4, Theorem 84]):

Theorem 1. Let 1 < p ^ 2, 0 < a ^ 1, f e Lip(a,p). Then f(t)e L3(R) for all ft satisfying the inequality

P ^ a ^ P < ft < Q =

p + ap — 1 p — 1

We will write that a non-negative measurable function X(t) e Ljoc(R+) belongs to the class A7, j ^ 1, if there exists C(j) ^ 1, such that

2 i+l 2i

l/7

( | A7(t)dtj 7 ^ C(-1 )2(1-7)/7 | X(t)dt, i e Z.

2

By the Holder inequality, it is easy to see that A71 c A72 for 1 ^ j2 < j1. It is proved in [8], that this embedding is strict. It is clear that a measurable function X(t) ^ 0 with the property

sup{A(i): 2* ^ t < 2i+1} ^ c inf{X(t): 2i-1 ^ t < 2i}, i e Z

is contained in all classes A7, 7 ^ 1. Further, we assume that X(t) = X(—t) for t > 0.

An analogue of (2) for sequences was introduced by Gogoladze and Meskhia [6]. The condition (2) was suggested by Moricz [13], who proved the following result:

Theorem 2. Let 1 <p ^ 2 and f e LP(R). If 1/p+ 1/q = 1, 0 <r < q,

and X e Ap/(p-rp+r), then

X(t)\f(t)\ dt ^ J X(t)t-r/^r(f, n/t)pdt. \t\>2 1

A more general result and the proof of its sharpness may be found in [8].

1

The aim of this paper is to obtain an analogue and generalization of Theorem 2 for the first Hankel-Clifford transform. Also, we estimate the rate of convergence of the corresponding integral. We generalize some results of Lahmadi, El Hamma and Mahfoud from [9] and [5]. Note that in [10] a less general results than in [5] are obtained. Some analogues of Theorem 1 for the Fourier-Bessel (or Hankel) transform were proved by Platonov [16]. Titchmarsh-type conditions for integrability of Fourier-Jacobi transforms and its generalization to Sobolev-Nikol'skii type spaces can be found in [4]. Analogues of Theorem 2 for Fourier-Dunkl and Fourier-Jacobi transforms can be found in [20] and [22].

2. Definitions. Let 1 ^ p <8, ^ ^ 0, and be the space of all

measurable real-valued functions with \\ f\\Lv = Q|/(x)\px^ dx) p <8. If

M 0

Xe is the indicator of a set E c R+ and f\E e Lp^(R+), then f e L^(E). By L°8(R+) we denote the space of bounded functions with the norm

\\f \\L8 = \\f \\® = SUp^eR+ \f (x^.

The Bessel-Clifford function of the first kind of order ^ ^ 0 (see, e.g., [7]) is defined by

-1)fc xk

C^(X) =

k-0

^ k\r(/i + k + 1)'

where r(a) is the Euler gamma function, and c^(x) is a solution of the differential equation

d2y , f , i\dy , n

dx2 dx

Let jv (x) be the normalized Bessel function of the first kind and order v > — 2, given by

> w-+1) Z 1) wr-

n=0 x '

Then c^ and are connected by

c^(x) = r-1^ + 1)j„(2?x), x > 0. (3)

If ^ ^ 0, then the first Hankel-Clifford transform h1,ll(f) is defined for appropriate functions f by

+co

h2,-{f)(y) = y" \ c-{Xx) f{x)dx 0

(see [12]). In [12] it is proved that the space H"{R+) consisting of all infinitely differentiable functions ^>{x) defined on R+, such that for all k,l e Z+ = {0,1,...}

sup

xldk{x "<p{x))

dxk

< 00

is invariant under the operator hi-.

Further we use the operator M^f{x) = x"f{x) and its inverse M-i. Since |C"{x)\ ^ r-i{| + 1) for x e R (see Lemma 2), the inequality

I I {■)-"hi-{/){■) I1 l* ^ r-V +1) I I f I I »" r-i{i +1) I I {-)-- f {■) I1 Li (4)

holds for/ e L"{R+), i.e., I IM- ihi-{f)I Il* < r-i{i +1) I IM- V IL. In [12] it is noted that in [11] several variants of Parseval-Plancherel equality are discussed. In particular, for f,g e L"{R+) one has

x "f {x)g{x) dx = y "Fi{y)Gi{y)dy

where Fi{y) = hl"{f){y), Gi{y) = hi,-{g){y). Whence, I I M~ %-{ f) 11 Li = 11 {-)-"hi"{/){■) 11 Li = 11 f {■) 11 Li = 11 M~ VI I Li . (5)

By the Riesz-Thorin interpolation theorem (see [2, Ch. 1, Theorem 1.1.1]), from (4) and (5) we deduce a Hausdorff-Young type inequality:

I I M" %,-{ f) 11 Li ^ C {p)I IM-7 hi, 1 ^P ^ 2, 1/p+ 1/q " 1, (6)

for f e L"{R+). Let A = A{x,y, z) be area of the triangle with sides x, y, z (A{x, y, z) = {p{p — x){p — y){p — z))i{2, where p = {x + y + z)/2). For i ^ 0, we set

A2"+i{x,y, z)

D"{x,y, z) =

2 2 " {xyz)-r{i + 2 )y/n

if the triangle with sides x, y, z exists, and D"{x, y, z) = 0 otherwise. Then D"{x,y, z) is non-negative and symmetric in x, y, z. In [17], Prasad, Singh, and Dixit introduced the generalized Hankel-Clifford translation of feL"{R+) by

+ 8

Tx(f)(y)= J f (z)D,(x,y,z)z"dz, 0 < X,y <8. 0

By Lemma 1.3 in [17], we have the following relation for f e L"(R+) between the first Hankel-Clifford transform and the generalized Hankel-Clifford translation:

hi,,(M,Tx(f))(t) = c^(xt)h^(MJ)(t), t > 0. (7)

The difference of order m e N with step t > 0 is

f (x) = (i - n^ + mr f (x).

We define the modulus of smoothness of order m e N in L"(R+) by

^m(f,S)P,^,hc = sup | | A^ f 11 Lp. o^t^s

The complicated form of equality (7) and inequality (6) obstruct to applying of differences of order m ^ 2 for h\^, e.g., there are doubts in the formula of Lemma 2.1 in [9].

Let \(t) be a non-negative measurable function from Ljoc(R+) and ¡j, ^ 0. If 7 ^ 1 and there exists C(j) ^ 1, such that

2y

I I M-1h1M) 11 Li " 11 (T^M )(■) 11 x7 (tr dt)1/7 ^

y

y

^C(1)y^+iqpi-7q/7 J X(t)t" dt, t > 0, (8)

y/2

then A e A7^.

Moricz [13] used similar conditions for y = 2% and 7 = 0 (see (2)), but in the proof of Theorem 1 it is more useful to apply (7).

3. Auxiliary propositions. From Lemma 1, we easily deduce the correctness of definitions of u1( f, #)pifJ,ihc.

Lemma 1. Let 1 ^p < 8, ^ ^ 0, f e Lp(R+). Then ||T(^ + 1)TtfI}LP ^

1 1 f11 L£ .

The proof of Lemma 1 can be found in [18, Lemma 1.1]

Lemma 2. Let i ^ 0. Then

(i) \j"{x)\ ^ 1 for x ^ 0 and j-{x) < 1 for x > 0;

(ii) 1 — j-{x) ^C > 0 for x ^ 1;

(iii) the inequality Cix2 ^ 1— j-{x) ^ C2x2 holds for some C2 > Ci > 0 and all x e [0,1].

Proof. The assertion of (i) can be found in [15], while the statement of (ii) is proved in [14, Lemma 3.3]. The right-hand inequality of (iii) is well known (see, e.g., [1]), the left-hand one is proved in [21] for 0 ^ x ^ y. By (i) and the continuity of j-{x), we also have the boundedness from below of {1 — 3"{x))/x2 on [q,1], and (iii) follows. □

Lemma 3. Let 1 ^ p ^ 2, | ^ 0, f e L"{R+), y ^ 0. Then hi,-{M-Tyf){x) = C"{yx)hi,"{M"f){x) a.e. on R+.

Proof. In the case p = 1, see (7). In particular, the formula of Lemma 3 is valid for f e S{R+). By definition, g{x) e S{R+), if the even extension of g to R belongs to the Schwartz space S{R). It is clear that S{R+) is dense in L"{R+). If fn e S{R+) and fn ^ f in L"{R+), then, by Lemma 1, Tyfn ^ Tyf in L"{R+), and, by (6) for 1 <p^2 and 1/p + 1/q = 1 or by (4) for p = 1 and q = 8, we have

limsup 11 M-i{hi-{ M-Tyfn) — hi,-{M-Ty f ))I I Li ^

^ Ci limsup IIM-iM-{Tyf — Tyfn)IIli = 0;

therefore,

0 = nlim 11 M-i{ c- {y)hi"{ M- fn) — hi"{ M-Ty f ))I Ili =

= 11 M-i{ c-{y)hi,-{ M-f) — hi,-{ M-Tyf ))I Ili ,

and the equality of Lemma 3 is proved a.e. on R+. □ 4. Main results.

Theorem 3. Let | ^ 0, 1 < p ^ 2, 1/p + 1/q = 1, f e LP-{R+). If X e Aq/pq-rqt- for some r e {0,q) and the integral

X{y)y-r/P-r/q ur { f,y-i)p^ihcy-dy N

converges for all N > 0, then \(t)\h1^(M^f)(t)\r e L*[N, + 8) for all N > 0 and

\(t)\h^(M^f )(t)\r rd t^

N

^Cj XPy)y*r/p-r/"pf,y-lU,hcy»dy. (9)

N/2

Proof. By Lemma 3 and the Hausdorff-Young-type inequality (6), we have:

y-q*\hAM*f )(t)\i(\ - J,(2?yt)ry*dy ^

8

£ Ci( Ux-*\M,A]^hcf(x)\)px*dxyP ^ C^mpf,t)p,,M.

, i/P 0

Let N > 0 be fixed and Di = [2N, 2i+1N), i e Z+. Taking U = 2-iN-1, by Lemma 2 (ii) we obtain

\hlAM*f)Py)\iy*dy ^

Di

^ C2P2iN r J y-*\hi,PMJ )py)\*pi - j*P2?yti)y y* dy ^

Di

^ C3p2NP f, . By the Holder inequality and the condition (8), we see that for 0 < r < q

XPy)\hi,*PM,f)Py)\ry*dy ^

< (J \Hy)\q/iq-rV dyy-r/q ^ \hh,PMJ)Py)\iy* dy)r'q ^

Di Di

^C,p2-iN-1 )p*+iy/^2iNy»u>1 Pf,UUM J \Py)y*dy^

Di-!

^C^ U>\ P f,y-1UMXy)y*r/p-r/q y* dy. (10)

Di-!

Di

Summing up (10) over i e Z+, we obtain (9). □

From Theorem 3 we deduce an integrability result on the whole R+. Theorem 4. Let | ^ 0, 1 < p ^ 2, 1/p + 1/q = 1, f e LP-{R+). If X e Aq/pq-rq>- for some r e {0, q), X e LqJPq r)[0,1) and the integral

X{y)y-r/p-r/q U>{ {f,y ^Uhhcy-dy (11)

converges, then X^i^M-f){t)\r e Ll-{R+). Proof. By (9), we find that

co

Xm^M-/){t)\r t-dt ^C j X{y)y-r /p-r/qu[ {f,y-i )p,ll,hcy-dy. (12) i i/2

By Lemma 1, we have {f,t)p,-,hc ^ C2IIf hn. Using the last inequality, the condition X e LqJPq r)[0,1) and the Holder inequality, we obtain

i I

J X{y)y-r/p-r/i{f,y-i)v,lthu^dy ^ CMTui^ X{y)y-dy ^

i/2 i/2

i i

^ C\X{y)\q/Pq-rV dy)1-r/q (J y-dy) r'q < 8 00

and both sides of (12) are finite. Finally, by the condition X e Lfq-r)[0,1) we see that

X^i^M-/){y)\r y-d y^

0

i i

,-di,\,q( \\1/(l-r)„,qr-/{q-r) ■■ - r/

^ y^hi,^ M- my^y^y) \X{y)\q/{q-r)y -dy) ^

00

i

i-r/q

^C5IIfHit q/pq-r)y-dy) - <8,

since 0 ^ yqr"(q r) ^ 1 for 0 ^ y ^ 1. Theorem is proved. □

Corollary 1. Let f, p, q, ^ and r are as in Theorem 4. If a > (r/q — 1)(fi + 1) and the integral

ya+r q/p-r/q^r ( f,y-1)p^My"dy (13)

converges, the ta\hl^(Mqf)(i)|r e Ll(R+).

Proof. It is easy to see that Xa(t) = ta belongs to Aq/(q-r),q for every

a e R. On the other hand, the condition Aa e LqJ (9-r)[0,1) is equiv-

1

alent to the convergence of integral $ tqa!(q-r)+q dt, or to the inequality

0

a > (r/q — 1)(fi + 1). Using Theorem 4, we obtain the statement of Corollary 1. □

Corollary 2. Let f, p, q, ^, and r are as in Theorem 4 and u1 (f, t)p>q>hc ^ Ct6 for some 8 > 0 and all t ^ 0. If a > (r/q — 1)(fi + 1), p8 + p > ^ + 1 and

p(a + ^ + 1) ^ A.

-p---)~<r< q, (14)

o p + p — ^ — 1

then ta\h1,q(Mqf)(t)\reL1q(R+).

Proof. Under conditions of Corollary 2, the integral (13) converges if a + rfi/p — r/q — rS + y < — 1 and r < q, i.e., r < q and r(1/q + 8 — ^/p) > a + ^ + 1. If 1/q + 8 — y/p = 8 + 1 — (¡i + 1)/p ^ 0, then r does not exist, while for p8 + p > ^ + 1 we obtain (14). □

Remark 1. The result of Corollary 2 in the case a = 0 coincides with Theorem 3 in [5]. In a similar manner, one can obtain the result of Theorem 4 in [5].

Acknowledgment. This work was supported by the Program of development of Regional Scientific and Educational Mathematical Center "Mathematics of Future Technologies" (project no. 075-02-2023-949).

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Received January 4, 2023. Accepted March 16, 2023. Published online March 24, 2023.

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